Type of artificial neural network suitable for learning and then predicting forest growth
My suggestion is not to use an ANN, but instead to use a simpler regression algorithm. The main reason for this is that ANNs take a long time to train, and work better when a lot of data is used. They also require a lot of expertise in parameter tuning to apply well. Since you say you don't have a lot of data, and also don't have a lot of experience using them, I think you will be better off applying something else first. If the other techniques don't work at all, then you might think about using ANNs, but again, they tend to want a lot of data.

If you have tried ordinary least squares regression, and found it does not work well, my next choice would be a Classification And Regression Tree. These models can make good decisions with small amounts of data, and do not require a lot of time to train. They can handle real-valued outputs like the height and width of a tree. Weka's REPTree might be a good starting place.

If Trees don't work out, my next suggestion would be to try regression using a Support Vector Machine. SciKitLearn's SVR is a good choice for this. SVRs can sometimes be very effective when data is limited, because they make assumptions about how to handle data-poor regions that seem to be generally applicable. An SVM can also report low confidence when estimating in those regions. They also train fairly fast when using small amounts of data, and can learn non-linear functions from the data.

If you really want to use an ANN, I would start with a simple Multi-layer perceptron. This model has few parameters to play with, and can probably fit well to your regression. It may make strange decisions in regions with less data however. 

Hope this helps!